#2038714

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1910 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6,9,10)(7,8,12,11)(21,22)(23,29)(24,26)(25,28)(27,30)(31,35)(32,33)(34,36);(6,10)(8,11)(13,14,16,18)(15,17,19,20)(22,24)(23,30)(25,33)(34,35);(5,6,9,10)(7,11,12,8)(13,14)(15,17)(16,18)(19,20)(21,23)(22,27)(24,29)(25,31)(26,30)(28,34)(32,35)(33,36);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(21,24,26,22)(23,29,30,27)(25,32,33,28)(31,35,36,34);(5,8,9,11)(6,7,10,12)(13,15,16,19)(14,17,18,20)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,35)(30,36)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=fg,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1945910);